Shape-type resonances

In the previous sections, we introduced resonance states and discussed situations in which resonances can be observed. In this section, we address the question of the origin for the appearance of resonances, or in other words, the basic question is what can bring about the formation of metastable states. In a very general manner, it is common to classify resonances into two main groups shape-type resonances and Feshbach-type resonances. Although the classification is not unique and may depend on the chosen representation of the Hamiltonian [46, 47], it can be extremely helpful in understanding the physical mechanism that leads to the formation of the metastable state. 

As the name suggests, shape-type resonances result from the shape of the potential at hand. But, what attributes must a potential have in order to trap the particle for a finite time and thus form a metastable state The wave nature of particles in quantum mechanics provides two typical ways for a 

In order to elucidate this concept, let us consider perhaps the simplest molecular system, the hydrogen molecular ion H2+. Within the Born-Oppenheimer approximation, the potential in the ground electronic state of this molecular ion is very well represented by the following Morse potential [48]  

Such high rotationally excited states in diatomic systems play a significant role in understanding molecular processes occurring in interstellar space [49]. For the specific system of H2+, these rotationally hot states can be produced for instance by the dissociation of CH42+ dications [50]. 

The situation depicted above is an example for the most common and vivid manner for the appearance of a resonance due to the shape of the potential. However, such metastable states can form even when the energy of the resonance state does not reside within some effective local well in the potential under study. A second way by which shape-type metastable states can form has much in common with optical resonators. In order to form a 

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Feshbach-type resonances [51], also known as Fano resonances [52] and Floquet resonances [22] depending on the system studied, are formed in a different manner. We encounter this type of metastable states whenever a bound system is coupled to an external continuum. In the same spirit as before, one can define a reference Hamiltonian in which the closed channel containing the bound states is uncoupled from the open channel through which the asymptote can be reached. When the coupling is introduced, the previously bound state decays into the continuum of the open channel. The distinction from shape-type resonances, described above, is that the resonance state decays into a different channel of the reference Hamiltonian. 

We will study, in particular, resonances that may occur for potentials that have a barrier over the threshold value separating an inner region from an outer region where the potential goes asymptotically to the threshold value. These resonances are known in the literature as shape-type [157] or potential [158] resonances. 

Transition metal electrodes prepared by sputtering and electrochemical deposition often show derivative-hke (i.e., bipolar) [17, 34—38] or negative absorption (anti-absorption) bands [46-50]. Spectral features change from normal absorption to anti-absorption through bipolar shapes with increasing amount of the metal deposited [17]. The bipolar band shape was ascribed to a Fano-type resonance of electronic interactions between molecular vibrations and the metal [34, 35]. The anti-... 

Fig. 7.Calculated Raman heterodyne signals of the detection beam showing Ramsey-type resonance line shapes. The sublevel detuning is normalized to the ground state relaxation rate Yg and the laser detuning is given in units of the Doppler width cte.Yvcc < notes the rate of velocity changing collisions.

Free radicals with resonance are definitely planar, though triphenylmethyl-type radicals are propeller shaped, like the analogous carbocations (p. 225). 

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